surfaces of constant curvature and their …...in § 4 another transformation is found which changes...

SURFACES OF CONSTANT CURVATURE AND THEIRTRANSFORMATIONS*

BY

LUTHER PFAHLER EISENHART

Bianchi f has called attention to the fact that upon a surface of positive total

curvature there is an infinity of conjugate systems of lines which are character-

ized by the property that when such a system is parametric the parameters can

be chosen so that the second quadratic form becomes

X(du2+dv2),

where X is generally a function of both u and v ; Bianchi calls such a system

isothermal-conjugate. In a similar manner it can be shown ^ that an infinity

of conjugate systems exist on a surface of negative curvature such that the

second quadratic form is reducible to

X(du2-dv2);

we call these also isothermal-conjugate. When the total curvature is constant

and positive, the surface is called spherical. We find that when a spherical

surface is referred to an isothermal-conjugate system the fundamental quantities

for the surface and the equations which they satisfy take very simple forms —

forms which suggest transformations carrying the given surface into new spher-

ical surfaces. This paper is given over to a study of these transformations.

In § 1 we derive the fundamental equations for a spherical surface referred to

isothermal-conjugate lines, show that its lines of curvature and characteristic

lines § are isothermal-conjugate, derive the expressions for the spherical repre-

sentation of the given isothermal system and for the fundamental functions

when this given system is composed of the lines of curvature or the character-

istic lines.

♦Presented to the Society April 29, 1905. Received for publication January 10, 1905, and

May 4, 1905.f Lezioni, vol. I, p. 167 ; German translation by Lukat, p. 135.

% Isothermal-conjugate systems of lines on surfaces, American Journal, vol. 25 (1903), p. 220.

§ Three particular systems oj ines on a surface, Transactions, vol. 5 (1904), p. 429.

472

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L. P. EISENHART : SURFACES OF CONSTANT CURVATURE 473

In § 2 we discover a transformation which changes a spherical surface into a

surface of the same curvature with the same second quadratic form as the given

surface and the lines of curvature are in correspondence on the two surfaces.

When the given surface is referred to its lines of curvature it is found that this

is the transformation of Hazzidakis. * The new transformation is the trans-

formation of Hazzidakis in a general form, for it can be shown that the latter

is the only transformation of spherical surfaces leaving the total curvature and

second quadratic forms unaltered and preserving lines of curvature.f

An interesting example of the effect of the Hazzidakis transformation is

afforded by applying it to the spherical surfaces of revolution. In addition to

this study § 2 is given over to the consideration of surfaces of constant mean

curvature, which are always associated with spherical surfaces, as Bonnet has

shown.

In § 4 another transformation is found which changes a spherical surface into

another, leaving the total curvature and the second quadratic form unaltered,

and changing lines of curvature into characteristic lines and vice versa. After

a study of the effects of combinations of these and Hazzidakis transformations,

it is shown that these transformations or transformations of the two kinds are

the only ones leaving the total curvature and second quadratic forms altered

and interchanging the lines of curvature and characteristic lines.

In § 5 the pseudospherical surfaces are considered and their equations of con-

dition are given forms analogous to those in § 1 for spherical surfaces, but one

cannot find transformations similar to the above.

§ 1. Spherical surfaces referred to isothermal-conjugate lines.

Consider a surface, S, of constant positive total curvature, 1/a2. Let the

parametric curves form an isothermal-conjugate system J upon the surface and

choose the parameters of these lines in such a way that

(1) D = D", Z>' = 0,

where these D's are the coefficients of the second quadratic form of the surface.

The Gauss § and Codazzi || equations can then be written in the respective

forms

«_*-*[¿(í{?})-¿(í{?])].

*BlANCHI, Lezioni, II, p. 436; German translation, p. 472.

t In a subsequent paper we shall show the part which the Hazzidakis transformation plays

in the theory of the deformation of surfaces.

% Blanchi, Lezioni, I, p. 167 ; German translation, p. 135.

¿Ibid., I, p. 93; p. 68.|| Ibid., I, p. 119 ; p. 91.


474 L. P. EISENHART : SURFACES OF CONSTANT [October

anddlogB [12 1 fill dlogB f 12 1 f 22

W Sr~ = 1 1 J ~ 1 2 [ ' ~~dT~ -1 2 } -11

where the Christoffel symbols {"} are formed with respect to the linear ele-

ment of S written

(4) ds2 = Edu2 + 2Fdudv + Gdv2,

and we have put for the sake of brevity

(5) H=VEG-F2.Since now

Bl 1

(6) F=ff2 = a2,we have

aiog^D _ dlogH Í 12 1 Í 22

~~dv~ ~ ~dv~ = j 1 J + ( 2

d log B d log H { 12 | [ 11

du ~ du ~ | 2 j T I 1

When these equations are combined with (3), the latter may be replaced by

9 — - — -™? 2d-F' -— dE^ ' du dv dv ' dv du du

From this it follows that the functions F and E — G are solutions of the equa-

tion

d20 d20

The general integral of this equation is

0= 4>(w + iv) + yt(u — iv),

where the functions 4> and ^ are arbitrary in form ; moreover, in order that

the surface be real SP must be conjugate to 4>. Consider this case and put

3>(w-f iv)+ ®0(u — iv)

2

when <1>0 denotes the conjugate function of <£> ; from (7) it follows that

E- G = i(<P-<í>0).

Since equations (7) have replaced the Codazzi equations, the only other equation

of condition to be satisfied is the Gauss equation (2). Hence <t> can be chosen


1905] CURVATURE AND THEIR TRANSFORMATIONS 475

arbitrarily and there remains a partial differential equation of the second older

to be satisfied by G; this equation is found by substituting the values for D2,

F, and E in (2).

As equation (8) is satisfied by 6 = 0, it follows that a solution is

(9) F=0, E- G= const.

Since the latter constant is zero only in case S is a sphere, we have the

theorem :

The lines of curvature upon a spherical surface form an isothermal-con-

jugate system ; these lines are isothermal-orthogonal only in case of the sphere.

Another solution of equations (7), suggested by the solution 6 = 0 of equation

(8), is

(9') E=G, /'= const.,

where the constant is equal to zero only for the sphere. Now

But the latter are the conditions that the conjugate parametric system is com-

posed of the lines cutting one another under the minimum angle between con-

jugate lines;* Reinaf has called the latter the characteristic lines. Hence

the theorem :

The characteristic lines upon a spherical surface are isothermal-conjugate.

From the general expressions for the coefficients of the linear element of the

spherical representation of a surface referred to a conjugate system, namely,4;

GD2 FDD' ED"2( ) ~EG-I2' ~~ EG-F2' <S~EG-F2'

it is seen that for the present case

G FE(ll) 5=a2' '"S»' § = a~2-

In consequence of (7) one has

d$ dS dg d& d§ dS

^ ' du dv dv ' dv du du '

that is, 6, &, § satisfy equations (7). Conversely, when S, &, § are three

functions satisfying (12), one has

* Three particular systems of lines on a surface, Transactions, vol. 5 (1904), pp. 9, 443.

t Dialcune proprietà delle linee caratlerische, Roma Accademia Lincei Rendiconti (4)

V (1899), pp. 881-5.

X Bianchi, Lezioni, vol. I, p. 150 ; German translation, p. 120.



{?}'♦{?!'"• {ÏM?}'"where the Christoffel symbols {\'} ' are formed with respect to

aV = Sdu2 + 2» dude + §dv2.

Hence if S, Sf, § satisfy also the Gauss equation for the unit sphere, the par-

ametric lines upon the sphere represent an isothermal-conjugate system of lines

on a unique spherical surface. *

§ 2. The transformation of Hazzidakis.

The results of the last section suggest a transformation of spherical surfaces.

Let S be such a surface with the linear element (4), whose coefficients satisfy

equations (7). We remark that the latter equations are satisfied also by Ex.

Fx, Gx, where

(13) EX=G, Ft=-F, GX = E.

If these values be subtituted in an equation similar to (2), namely,

an *-«{¿(?{Y})-£(5l?})].when the Christoffel symbols are formed with respect to

ds\ = Exdu2 + 2Fldudc + Gxdv2,

and one remarks that

Hx = II,

it follows in consequence of (7) that this becomes

«"[£(S{?}K(i{?})].where now the symbols are formed with respect to (4). But the right-hand

member of this equation is equivalent to the similar member of (2).f Hence,

the Gauss equation is satisfied by taking Bx = B'x = B. This gives the

theorem :

When a spherical surface is referred to an isothermal-conjugate system and

the parameters are chosen so that (1) is satisfied, there exists another spherical

surface of the same total curvature and with the same second quadratic form

as the given surfaces ; the linear element is

(14) ds2 = Gdu2 - 2Fdudv + Edv2.

•Ibid., p. 168; p. 136.

tlbid., p. 77; p. 53.



It will be convenient to refer to the new surface Sx as the conjugate of the

given one. One remarks that the relation between the two surfaces is entirely

reciprocal.

If the linear element of the spherical representation of Sx is written

dcr\ = sxdu2 + 2ffxdudv +§xdv\

it follows from (10), (11) and (13) that

F(15) sx = -2=§, *i--#, §x=s.

Hence the coefficients of the spherical representation of S and Sx bear the same

relation to one another as the first fundamental quantities of these surfaces.

A comparison of the formulas (11), (13) and (15) leads to the theorem:

A spherical surface of unit total curvature is applicable to the unit sphere

'in such a way that an isothermal-conjugate system of lines on the former cor-

respond to the spherical representation of the corresponding isothermal-conju-

gate system on its conjugate.

From the general equation of the lines of curvature

( ED' ■- FD)du2 + (ED" - GD)dudv + (FD' -GD')dv2 = 0,

it is seen that for both S and Sx the lines of curvature are given by

(16) F du2 + (G - E)dudv - Fdv2 = 0.

Hence the lines of curvature correspond upon a spherical surface and its

conjugate.

Suppose that the isothermal-conjugate lines on S are the lines of curvature :

from (9) it is seen that we can put

(17) E=a2eosh26, G = a2 sinh2 0, F=0

or

(18) E=a2sinh20, G = a2 cosh2 0, F=0

according as the constant in (9) is positive or negative, a2 being the absolute

value of the constant.* Now in either case

D = D" = a sinh 6 cosh 0,

and equation (2) becomes

d20 d20 .,„,„„-s-a 4- -tï + sinh 0 cosh 0 = 0.du2 dv'

•Lezioni, vol. II., p. 436 ; p. 472.



One sees that the values (18) lead to the conjugate of the surface whose first

fundamental coefficients are given by (17) and vice versa. But this is the trans-

formation which Bianchi has called the transformation of Hazzidakis.

In order to show that the transformation which we have been considering in

this section is the transformation of Hazzidakis in a more general form, we

have only to prove the theorem :

The transformation of Hazzidakis is the only one changing a- spherical

surface into one of the same curvature in such a way that the lines of curva-

ture on the two surfaces correspiond and the second quadratic forms are the

same.

Let S and Sx he two surfaces of curvature 1/a2 referred to their correspond-

ing lines of curvature and let the second quadratic forms of the two surfaces be

(19) B(du2+dv2).

If no distinction is made with regard to homothetic surfaces, the most general

integrals of equations (7) for S and Sx are

(20) E- G=±a2, Ex- Gx = ±a2.

Since the total curvatures of S and Sx are equal and (19) is the second quad-

ratic form of both surfaces, we must have also

(21) EXGX = EG.In conformity with this we put

EX=XE, GX = \G.X

Substituting these values in the second of equations (20) and taking account of

the first, we get

\2E=f a2X + (±a2-E) = 0,

of which the solutions are

GX=±l, ±t¿.E

Since Ex and Gx must be positive, the only possibility is

Ex = G, Gx = E,

which proves the theorem. Hence the Hazzidakis transform, or conjugate, of a

spherical surface may be defined to be the surface with the same total curvature

and second quadratic form and such that the lines of curvature on the two sur-

faces are in correspondence.

We shall find that on the two surfaces S and Sx the characteristic lines also



are in correspondence. For, the general equation of the characteristic lines upon

a surface referred to a conjugate system is *

D(GD- ED")du2 - IFDD'dudv + D"(ED" - GD)dv2 = 0.

When the system is such that the second quadratic form is (19), this reduces to

(22) (G-E)du2- IFdudv + (E-G)dv2 = 0,

which in consequence of (13) is the equation of these lines on Sx also. Hence

the characteristic lines correspond upon two conjugate spherical surfaces.

§ 3. Surfaces of revolution and surfaces of constant mean curvature.

The spherical surfaces of revolution, referred to their meridians and parallels,

lend themselves readily to the transformation of Hazzidakis. Bianchi f shows

that these surfaces are defined by the equations

(23) x = r cos w, y = rsinw, z = (b(r),

where

(24) r=KC0su, z=J \/l — K2 sin2 u du,

in which k denotes a constant. When k is equal to one, S is a sphere of unit

radius. The other surfaces of this kind are divided into two classes according

as k is less or greater than one. From (23) and (24) it is found that the linear

element of S has the formdr2

(25) ds2=-^--^ + r2dw2

and for the second quadratic form we have

(26) *~^l-«8 + 4(^)(l-«2 + ^-) + ^2]-

As the case of the sphere does not afford any particular interest, we pass to the

consideration of the two remaining cases in turn.

Io. k < 1. In order to have D = D" we change the parameters, putting

drdt =

1/(l_/e2+r^(/c2_r2)

Upon integration we have r = k cnt, with k the modulus. When this change is

effected upon (25) and (26), they become

* Three particular systems of lines, 1. c, p. 429.

t L. c, vol. 1, p. 233 ; German translation, p. 197.


480 L. p. eisenhart: surfaces of constant [October

(27) oV = dn21 dt2 + k2 en21 dw2

(28) * = K cnt dnt ( df + dw2 ).

2°. kx> 1. For this case we put

dt dr

*i = i/(l-«J + A)(*2 - r-) '

which gives by integration r2 = k\ — sn2t, and the modulus is l/tcx. Now the

linear element and second quadratic form, after patting u\ = kxw, reduce to

{*?} t(29) ds2=- 2-d€- + dn-tdw\,

Ki

„„. ^ cnt-dnt, , „ , .,(30) 4> = ((/^+*y).

If we take for kx the inverse of tc, the elliptic functions have the same modu-

lus. Again, if we take the same t for both cases and to a line w = const, on

one of the first case make correspond a line wx = const, on a surface of the

second kind, the forms (28) and (30) are the same, and (30) is obtained by

interchanging the coefficients in (28) and vice versa. Hence :

Given a surface of revolution of constant positive curvature corresponding

to a certain k; its conjugate is the similar surface corresponding to 1/tc.

Bonnet has shown that there is a surface of constant mean curvature l/2a

parallel to a surface of total curvature 1/a2 and at the distance a. If the para-

metric curves be the lines of curvature for these surfaces and the fundamental

functions for the former surface be denoted by e, g, d, d", the following

relations hold : *

-»-v,[q(ft+l)-']'

where

X = d = d".

Let S and Sx he two surfaces of unit total curvature which are the trans-

forms of one another and denote by S and Sx the two surfaces of constant mean

curvature respectively which are associated with the former, as remarked by the

theorem of Bonnet. Now a in the above formulae has the value unity. If the

functions for Sx he denoted by Ex, Gx, Bx, D"x, the above relations lead to

the following in consequence of the relation between S and Sx :

e = g = c, = a,, d = d'i, d" = dx.

* Knoblauch, Einleitung in die allgemeine Theorie der krummen Flachen, p. 234.



We have then the theorem :

Given any surface of constant mean curvature, there is another surface of

this kind applicable to it in such a way that lines of curvature correspond ;

when the latter are parametric and the parameters are suitably chosen

D = D'f D" = DX.

Bonnet * has shown that every surface of constant mean curvature is appli-

cable to an infinity of surfaces of the same mean curvature with preservation of

the principal radii. It has just been seen that to every surface of this kind

there is applicable a surface of the same mean curvature with preservation of

the lines of curvature and equality of the principal radii in inverse order.

Hence the theorem :

Given a surface S of constant mean curvature, there are two families of

surfaces of the same mean curvature applicable to it ; for the surfaces of the

one family the principal radii are equal to the corresponding radii of S, and

for the surfaces of the other family the principal radii are equal to those of S

in reverse order.

§ 4. A new transformation of spherical surfaces.

Let E, F, G be three functions satisfying equations (7). These equations

are satisfied also by

«11 F E+G-2F E-G E+G + 2F(6x) Hx= — 2 ,jfv,= -£— , O, = -2-.

We find that

(32) EXGX-F\ = EG-F2,

so that we are led to inquire whether there exists a surface Sx whose first funda-

mental coefficients are given by (31) and whose second quadratic form is the

same as for S. In order that this be true, it is necessary and sufficient that

these functions satisfy the Gauss equation (13'), which in consequence of (6),

(31) and (32) can be given the form

(33)

id d (E — G\ d r 1 d 14//a2={^log(/;+(?-2/')att(-^-)-5M[//ÔM^+i? + 2/')J

-Í [TOW G+2Fï] - l^ <*+ *~**) ¡v (HÈr) } •

* Mémoire su r la théorie des sur]'aces applicables sur une surface donnée, Journal de l'École

Polytechnique, Cahier 42, p. 77.



The Gauss equation for S can be reduced to either of the equivalent forms

(34) 2a2 H= j= -fofoyjfj - du\ff~du~) ~dv\H^iT)~ E~du dv \HJ'

(ÓO) ¿a Jl- G dudv ^jjj dv\Hdv ) du\Hdu) G dv du\Hf

If these equations be added and the result be subtracted from (33), we will be

brought to an equation which vanishes identically in consequence of (7). Hence :

Given a spherical surface referred to an isothermal-conjugate system of lines

and let the parameters be so chosen that the second fundamental functions D

and D" are equal ; there exists a second spherical surface with the same second

quadratic form- and the coefficients of its linear element are given by (31). For

convenience we shall call the new surface Sx the adjoint of S. When the ex-

pressions (31) are substituted in the equation of the characteristic directions on

£,, namely (22)

( Gx - Ex)du2 - 4Fxdudv + (Ex -Gx)dv2 = 0,

this equation reduces to (16). Hence the characteristic lines on Sx correspond

to the lines of curvature on S. And the equation of the lines of curvature on

Sx reduces to (22). Hence upon a spherical surface and its adjoint the lines

of curvature of one surface correspond to the characteristic lines upon the

other.

For the spherical representation of Sx we have

Gx - Fx Exêl==~c?' *í=-^-' &-¿ar>

which in consequence of (31) and (11) may be written

S+§ — 2& S — § S + § + 2&sx= 2 ' ^= ~~2~' ^l = 2~~ ~'

Hence the coefficients of the spherical representation of corresponding isother-

mal-conjugate lines on a spherical surface and its adjoint bear the same rela-

tion to one another as the first fundamental functions for these surfaces.

For the sake of convenience we shall refer to the transformation of Hazzi-

dakis and the one treated in the preceding section as the transformations H

and / respectively. First of all we remark that throughout any number of

operations of H and / separately or conjointly the second quadratic form is

invariant. Since the relation between a surface and its transform by either H

or / is completely reciprocal, it follow from (13) and (31) that the same sur-



face is obtained when H and I are applied consecutively as when their order is

reversed. And also that one is brought back to the original surface by any one

of the sets of combined transformations

HH, HII, IHI, IIH.

Denote by T any transformation of a spherical surface into a second surface

of this kind which preserves the total curvature and second quadratic form

and interchanges lines of curvature and characteristic lines. Then IT leaves

lines of curvature unaltered and consequently is either identity or an H trans-

formation. Therefore T is either an I transformation or a combination of an

I and one or more H transformations. Hence the theorem :

The only transformation which changes a spherical surface into a spherical

surface of the same curvature and the same second quadratic form and inter-

changes the lines of curvature and characteristic lines is the I transformation

or a combination of the latter and Hazzidakis transformations.

§ 5. Surfaces of negative curvature.

In this section we study surfaces of negative curvature in a manner similar to

the preceding. We have shown * that upon such a surface there is an infinity

of conjugate systems such that, when these lines are taken as parametric, the

parameters can be so chosen that

B=-B", B' = 0.

The Gauss equation can be written

*-¿K(í{?})-a(§{Y})].and proceeding as in § 1 we find that if we put

B2

(37) F=-Jf2=-1,

the Codazzi equations can be put in the form

m) 2-^= — + ^ 2d^=Ô^+dG-.^ ' du dv dv ' dv du du

From this it is seen that F and E + G are solutions of

d20 d20

<39) duJ-dW = °-

* Isothermal-conjugate Systems, Amerioan Journal of Mathematics, vol. 25 (1903), p.

220.

Trans. Am. Math. Soc. 38License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


When either of these functions is known, the other is given by a quadrature

from (38) and then it remains to find E or G from (36). This shows that

every solution of (39) leads to surfaces of constant negative curvature. The

most general form which F can have is

®(u + v) + V(u — v)A - - -g- —,

and thenE + G=®(u + v) — V(u — v).

It is clear that equations (38) are satisfied by

(40) EX = G, GX = E, FX = F.

When these values of A", G, F are substituted in (36) with F replaced by — 1,

and it is noted that they satisfy (38), we get

where the Christoffel symbols { " }x are formed with respect to

(41) Ex du2 + 2FX du dv + Gx dv2.

But the left-hand side of the above expression is equal to the total curvature of

the surface with the linear element (41)* Hence when these coefficients have

the values (40), the surface Sx has positive curvature, so that for pseudospher-

ical surfaces a transformation analogous to the transformation of Hazzidakis

does not exist. The linear element of the spherical representation of S is readily

found to be fda2 = Gdu2 + 2Fdudv + Edv2,

which is the same as (41) when the coefficients in the latter have the values

given by (40) ; this accounts for the above result.

It is of interest to determine whether there is a surface with total curvature

equal to one whose linear element takes this form when the parametric lines

form an isothermal conjugate system. The conjugate of this surface would have

for linear element

ds2 = Edu2 - 2Fdudv + Gdv2,

so that E, F, G would have to satisfy equations

2 ÔF - — - d~ 2 ----- -~

\ ' du dv dv ' dv du du

* Bianchi, 1. c., vol. I., p. 77; German translation, p. 53.

t Ibid, p. 150 ; p. 120.



as well as (38). For these to be satisfied it is necessary that E be a function of

m alone and G of v alone. In consequence of (42), we have for the forms of

these functions

E = au2 + 2yu + e, F = auv + ßu + yo + 8, G = av2 + 2ßv + k,

where a, ß, y, 8, e and « are constants. For no values of these constants does

the formula (36) make K equal to — 1, which shows the non-existence of such

a transformation.

One sees also that if E, G, Fare solutions of equations (38) so also are

E-G + 2F G-E+2F E+GA =-2-' "" 2 ' *-2~~'

But now

EXGX-F\=-(EG-F2);

hence the transform is imaginary.

Princeton University.


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